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I'm looking for a quick approach to a rigorous proof of the central limit theorem of probability theory.

(I imagine that this would have to be a paper or monograph with this as its main goal, although, conceivably, a textbook on probability may give a reasonably quick approach.)

I realize that there are many versions of the central limit theorem. It is my understanding that it is Lyapunov's version that is considered the most general of them, so this would be the one I'm most interested in (though I'm open to suggestions on this question).

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Lindeberg version is more general I think. I believe every proof of the CLT goes through characteristic functions so I don't think you'll manage to find an easier approach. as it's very standard theorem I believe you'll find it in pretty much every textbook on probability for university students, try Durrett's probability theory or Feller's introduction to probability for example – mm-aops Jun 25 '13 at 13:31
Thanks for pointing out Lindeberg's version, and for the pointer to Durrett's book (remarkable agreement with nrpeterson's suggestions). Feller's classic has many merits, but it's anything but quick... – kjo Jun 25 '13 at 14:05

Actually, Lindeberg's CLT is slightly stronger than Lyapunov's; the assumption needed to prove it is slightly weaker.

I learned the CLT out of Durrett:

R. Durrett. Probability: Theory and Examples. Duxbury, Belmont, CA, 3rd edition, 2005.

That text has a great chapter devoted entirely to the development of the necessary tools: weak convergence, characteristic functions, etc. I believe he refers to Lindeberg's CLT as the Lindeberg-Feller Theorem.

If you don't have access to the text, Durrett himself has posted "edition 4.1" of his book as a pdf on his website.

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Thanks for the pointer to Durrett, and especially the link! I'll check it out. – kjo Jun 25 '13 at 14:06

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